Make the biscuits: In a medium bowl, mix the flour, granulated sugar, baking powder, baking soda and salt. Using a pastry blender or 2 knives, cut in the butter until it is the size of peas. Gently stir in the buttermilk until the dough just comes together.
Transfer the dough to a lightly floured surface and pat it into a 3/4-inch-thick disk. Using a 2 1/2-inch round biscuit cutter, stamp out 8 biscuits, patting the scraps together as needed. Transfer the biscuits to a parchment paper–lined baking sheet. Brush the tops with the cream and sprinkle with the turbinado sugar. Refrigerate the biscuits until chilled, about 30 minutes.
Preheat the oven to 450°. Bake the biscuits for about 15 minutes, until lightly golden; transfer to a rack to cool. Reduce the oven temperature to 375°.
Make the cobbler: Lightly butter an 8 1/2-by-11-inch baking dish. Spread the pecans in a pie plate and toast in the oven for 8 minutes, until fragrant; let cool, then coarsely chop. In a bowl, toss the peaches, brown sugar, granulated sugar, cornstarch, cinnamon and vanilla. Fold in the pecans.
Spread the peaches in the prepared baking dish and cover with foil. Set the dish on a rimmed baking sheet and bake for 20 minutes, until the peaches begin to release their juices. Remove the foil and bake for about 30 minutes longer, stirring once, until the peaches are bubbling. Arrange the biscuits over the fruit. Bake the cobbler for 5 minutes longer. Transfer to a rack to cool slightly.
Meanwhile, make the whipped cream: In a large bowl, combine all of the ingredients. Using an electric mixer, beat the cream at medium speed until softly whipped. Spoon the cobbler into shallow bowls, top with dollops of whipped cream and serve.